# Disjunctive Representation of Triangular Bipolar Neutrosophic Numbers, De-Bipolarization Technique and Application in Multi-Criteria Decision-Making Problems

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## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

#### 1.2. Novelties

#### 1.3. Verbal Phrasesin theNeutrosophic Arena

**Example**

**1.**

#### 1.4. Logical Relationship between the Objective and the Subjective Partsof this Paper

#### 1.5. Structure of this Paper

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

## 3. Single Typed Linear Triangular Bipolar Neutrosophic Number

#### 3.1. Triangular Single Typed Bipolar NeutrosophicNumber of Category-1: The Portion of the Authenticity, Hesitation, and Untrue Are Independent

#### 3.2. Triangular Single TypedBipolar Neutrosophic Number of Category-2: The Portion of Hesitation and Untrue Are Dependent

#### 3.3. Triangular Single TypedBipolar Neutrosophic Number of Category-3: The Portion of the Authenticity, Hesitation, and Untrue Are Dependent

## 4. Single Typed Nonlinear Triangular Bipolar Neutrosophic Number

#### 4.1. Single Typed Nonlinear Triangular Bipolar Neutrosophic Number

#### 4.2. Single Typed Generalized Triangular Bipolar Neutrosophic Number

#### 4.3. Single Typed Generalized Non Linear Triangular Bipolar Neutrosophic Number

## 5. De-Bipolarization of a Linear Neutrosophic Triangular Bipolar Fuzzy Number

- BADD (basic defuzzification distributions)
- BOA (bisector of area)
- CDD (constraint decision defuzzification)
- COA (center of area)
- COG (center of gravity)
- ECOA (extended center of area)
- EQM (extended quality method)
- FCD (fuzzy clustering defuzzification), etc.

#### 5.1. De-Bipolarization Using the Removal Area Method

## 6. Multi-Criteria Decision-Making in a Triangular Bipolar Neutrosophic Fuzzy Set Environment

#### 6.1. Illustration of the MCDM Problem

#### 6.2. Weighted Mean and Normalisation Algorithm of the MCDM Problem

#### 6.3. Illustrative Example

#### 6.4. Results and Sensitivity Analysis

#### 6.5. Comparison with Other Established Work:

**Remark**

**1.**

## 7. Conclusions and Future Research Scope

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Block diagram for different types of a single typed linear triangular bipolar neutrosophic number.

**Figure 4.**(

**a**) Shaded Region of falsity portion (Step I); (

**b**) Shaded Region of falsity portion (Step II).

**Figure 5.**(

**a**) Shaded Region of hesitant portion (Step I); (

**b**) Shaded Region of hesitant portion (Step II).

**Figure 6.**(

**a**) Shaded Region of truth portion (Step I); (

**b**) Shaded Region of truth portion (Step II).

Distinct Parameter | Verbal Phrase | Information |
---|---|---|

Interval Number | [Low, High] | Voter will select according to their first priority within a certain range, like [2nd,3rd] candidate. |

Triangular Fuzzy Number | [Low, Median, High] | Voter will select according to their first priority containing an intermediate candidate, like [1st,2nd,3rd] |

Intuitionistic (Triangular) | [Standard, Median, High; Very Low, Poor, Low] | Voters will select some candidates directly and reject others immediately according to their viewpoint. |

Neutrosophic (Triangular Bipolar) | [High, Standard, Very High; Intermediate, Average, Median; Very Low, Poor, Low] | Some voters will select some candidates directly, some will hesitate when casting their vote, and some will directly reject voting according to their viewpoint. |

Cases | Attribute | Verbal Phrase |
---|---|---|

Quantitative Attributes | ||

1 | Price of the product | Very high (VH), High (H), Intermediate (I), Small (S), Very small (VS) |

2 | Legibility of the product | Very high (VH), High (H), Mid (M), Low (L), Very low (VL) |

3 | Service of the product | Very high (VH), High (H), Mid (M), Low (L), Very low (VL) |

Alternatives/Attributes | C^{1} | C^{2} | C^{3} |
---|---|---|---|

A^{1} | L | M | H |

A^{2} | VL | M | I |

A^{3} | L | I | VH |

Attribute Weight | Final Decision Matrix | Ordering |
---|---|---|

<(0.33,0.30,0.37> | $\left(\begin{array}{c}<6.983>\\ <7.22>\\ <6.79>\end{array}\right)$ | ${P}_{2}>{P}_{1}>{P}_{3}$ |

<(0.25,0.30,0.45> | $\left(\begin{array}{c}<6.87>\\ <7.37>\\ <6.95>\end{array}\right)$ | ${P}_{2}>{P}_{3}>{P}_{1}$ |

<(0.35,0.25,0.40> | $\left(\begin{array}{c}<6.85>\\ <7.15>\\ <6.68>\end{array}\right)$ | ${P}_{2}>{P}_{1}>{P}_{3}$ |

<(0.40,0.30,0.30> | $\left(\begin{array}{c}<7.04>\\ <7.18>\\ <6.58>\end{array}\right)$ | ${P}_{2}>{P}_{1}>{P}_{3}$ |

<(0.20,0.30,0.50> | $\left(\begin{array}{c}<6.82>\\ <7.24>\\ <7.03>\end{array}\right)$ | ${P}_{2}>{P}_{3}>{P}_{1}$ |

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**MDPI and ACS Style**

Chakraborty, A.; Mondal, S.P.; Alam, S.; Ahmadian, A.; Senu, N.; De, D.; Salahshour, S.
Disjunctive Representation of Triangular Bipolar Neutrosophic Numbers, De-Bipolarization Technique and Application in Multi-Criteria Decision-Making Problems. *Symmetry* **2019**, *11*, 932.
https://doi.org/10.3390/sym11070932

**AMA Style**

Chakraborty A, Mondal SP, Alam S, Ahmadian A, Senu N, De D, Salahshour S.
Disjunctive Representation of Triangular Bipolar Neutrosophic Numbers, De-Bipolarization Technique and Application in Multi-Criteria Decision-Making Problems. *Symmetry*. 2019; 11(7):932.
https://doi.org/10.3390/sym11070932

**Chicago/Turabian Style**

Chakraborty, Avishek, Sankar Prasad Mondal, Shariful Alam, Ali Ahmadian, Norazak Senu, Debashis De, and Soheil Salahshour.
2019. "Disjunctive Representation of Triangular Bipolar Neutrosophic Numbers, De-Bipolarization Technique and Application in Multi-Criteria Decision-Making Problems" *Symmetry* 11, no. 7: 932.
https://doi.org/10.3390/sym11070932